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General articles

Construction of uniformly symmetric bi-frames based on multiresolution template algorithm

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Pages 1192-1216 | Received 27 Feb 2017, Accepted 25 Feb 2018, Published online: 20 Mar 2018

References

  • M. Bertram, Biorthogonal loop-subdivision wavelets, Computing 72 (2004), pp. 29–39. doi: 10.1007/s00607-003-0044-0
  • M. Bertram, M.A. Duchaineau, B. Hamann, and K.I. Joy, Generalized B-spline subdivision-surface wavelets for geometry compression, IEEE Trans. Vis. Comput. Graph. 10 (2004), pp. 326–338. doi: 10.1109/TVCG.2004.1272731
  • J.F. Cai, R.H. Chan, L.X. Shen, and Z.W. Shen, Restoration of chopped and nodded images by framelets, SIAM J. Sci. Comput. 24 (2008), pp. 1205–1227. doi: 10.1137/040615298
  • J.F. Cai, R.H. Chan, and Z.W. Shen, A framelet-based image inpainting algorithm, Appl. Comput. Harmon. Anal. 24 (2008), pp. 131–149. doi: 10.1016/j.acha.2007.10.002
  • J.F. Cai, S. Osher, and Z.W. Shen, Split Bregman methods and frame based image restoration, Multiscale Model. Simul. 8 (2009), pp. 337–369. doi: 10.1137/090753504
  • C.K. Chui, W.J. He, and J. Stöckler, Compactly supported tight and sibling frames with maximum vanishing moments, Appl. Comput. Harmon. Anal. 13 (2002), pp. 224–262. doi: 10.1016/S1063-5203(02)00510-9
  • I. Daubechies and B. Han, The canonical dual frame of a wavelet frame, Appl. Comput. Harmon. Anal. 12 (2002), pp. 269–285. doi: 10.1006/acha.2002.0381
  • I. Daubechies and B. Han, Pairs of dual wavelet frames from any two refinable functions, Constr. Approx. 20 (2004), pp. 325–352. doi: 10.1007/s00365-004-0567-4
  • I. Daubechies, B. Han, A. Ron, and Z.W. Shen, Framelets: Mra-based constructions of wavelet frames, Appl. Comput. Harmon. Anal. 14 (2003), pp. 1–46. doi: 10.1016/S1063-5203(02)00511-0
  • B. Dong and Z.W. Shen, Pseudo-splines, wavelets and framelets, Appl. Comput. Harmon. Anal. 22 (2007), pp. 78–104. doi: 10.1016/j.acha.2006.04.008
  • B. Dong and Z.W. Shen, MRA-based wavelet frames and applications, in Mathematics in Image Processing, Z. Hong-Kai, ed., IAS Lecture Notes Series, Vol. 19, American Mathematical Society, 2013.
  • B. Dong, Q.T. Jiang, and Z.W. Shen, Image restoration: Wavelet frame shrinkage, nonlinear evolution PDEs, and beyond, Multiscale Model. Simul. 15 (2017), pp. 606–660. doi: 10.1137/15M1037457
  • M. Ehler, On multivariate compactly supported bi-frames, J. Fourier Anal. Appl. 13 (2007), pp. 511–532. doi: 10.1007/s00041-006-6021-1
  • M. Ehler and B. Han, Wavelet bi-frames with few generators from multivariate refinable functions, Appl. Comput. Harmon. Anal. 25 (2008), pp. 407–414. doi: 10.1016/j.acha.2008.04.003
  • B. Han, On dual wavelet tight frames, Appl. Comput. Harmon. Anal. 4 (1997), pp. 380–413. doi: 10.1006/acha.1997.0217
  • B. Han, Computing the smoothness exponent of a symmetric multivariate refinable function, SIAM J. Matrix Anal. Appl. 24 (2003), pp. 693–714. doi: 10.1137/S0895479801390868
  • B. Han, Dual multiwavelet frames with high balancing order and compact fast frame transform, Appl. Comput. Harmon. Anal. 26 (2009), pp. 14–42. doi: 10.1016/j.acha.2008.01.002
  • B. Han, The structure of balanced multivariate biorthogonal multiwavelets and dual multiframelets, Math. Comput. 79 (2010), pp. 917–951. doi: 10.1090/S0025-5718-09-02320-5
  • B. Han, Algorithm for constructing symmetric dual framelet filter banks, Math. Comput. 84 (2015), pp. 767–801. doi: 10.1090/S0025-5718-2014-02856-1
  • B. Han and Z.W. Shen, Dual wavelet frames and Riesz bases in Sobolev spaces, Constr. Approx. 29 (2009), pp. 369–406. doi: 10.1007/s00365-008-9027-x
  • Q.T. Jiang, Orthogonal and biorthogonal 3-refinement wavelets for hexagonal data processing, IEEE. Trans. Signal. Process. 57 (2009), pp. 4304–4313. doi: 10.1109/TSP.2009.2026538
  • Q.T. Jiang, Bi-frames with 4-fold axial symmetry for quadrilateral surface multiresolution processing, J. Comput. Appl. Math. 234 (2010), pp. 3303–3325. doi: 10.1016/j.cam.2010.04.029
  • Q.T. Jiang, Wavelet bi-frames with uniform symmetry for curve multiresolution processing, J. Comput. Appl. Math. 235 (2011), pp. 1653–1675. doi: 10.1016/j.cam.2010.09.008
  • Q.T. Jiang, Biorthogonal wavelets with 4-fold axial symmetry for quadrilateral surface multiresolution processing, Adv. Comput. Math. 34 (2011), pp. 127–165. doi: 10.1007/s10444-009-9144-5
  • Q.T. Jiang, Biorthogonal wavelets with 6-fold axial symmetry for hexagonal data and triangle surface multiresolution processing, Int. J. Wavelets. Multiresolut. Inf. Process. 9 (2011), pp. 773–812. doi: 10.1142/S0219691311004316
  • Q.T. Jiang, Correspondence between frame shrinkage and high-order nonlinear diffusion, Appl. Numer. Math. 62 (2012), pp. 51–66. doi: 10.1016/j.apnum.2011.10.002
  • Q.T. Jiang and P. Oswald, Triangular 3-subdivision schemes: The regular case, J. Comput. Appl. Math. 156 (2003), pp. 47–75. doi: 10.1016/S0377-0427(02)00904-4
  • Q.T. Jiang and D.K. Pounds, Highly symmetric bi-frames for triangle surface multiresolution processing, Appl. Comput. Harmon. Anal. 31 (2011), pp. 370–391. doi: 10.1016/j.acha.2011.01.007
  • Q.T. Jiang and D.K. Pounds, Highly symmetric 3-refinement biframes for surface multiresolution processing, Appl. Numer. Math. 118 (2017), pp. 1–18. doi: 10.1016/j.apnum.2017.02.005
  • A. Ron and Z.W. Shen, Affine systems in L2(Rd): The analysis of the analysis operator, J. Funct. Anal. 148 (1997), pp. 408–447. doi: 10.1006/jfan.1996.3079
  • A. Ron and Z.W. Shen, Affine systems in L2(Rd): Dual systems, J. Fourier Anal. Appl. 3 (1997), pp. 617–637. doi: 10.1007/BF02648888
  • Z.W. Shen, Wavelet frames and image restorations, Proceedings of the International Congress of Mathematicians, Hyderabad, India, 2010.
  • W. Sweldens, The lifting scheme: A custom-design construction of biorthogonal wavelets, Appl. Comput. Harmon. Anal. 3 (1996), pp. 186–200. doi: 10.1006/acha.1996.0015
  • H.W. Wang, K.H. Qin, and K. Tang, Efficient wavelet construction with Catmull–Clark subdivision, Vis. Comput. 22 (2006), pp. 874–884. doi: 10.1007/s00371-006-0074-7
  • H.W. Wang, K.H. Qin, and H.Q. Sun, 3 -subdivision-based biorthogonal wavelets, IEEE Trans. Vis. Comput. Graph. 13 (2007), pp. 914–925. doi: 10.1109/TVCG.2007.1031

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